Greco-Roman wrestling: One of the oldest and most widely practiced forms of wrestling in the world, modern Greco-Roman wrestling is an Olympic sport that traces its origins to the days of France during the Napoleonic era. One of the most notable rules in the Greco-Roman style is the restriction on grabs below the waist, which places a premium on a competitor’s ability to throw his opponent to the ground. Common moves found in Greco-Roman wrestling include headlocks, bear hugs, and arm drags.
Freestyle wrestling: Another wrestling style that takes place in the Olympics, freestyle wrestling as an organized sport traces its roots to the days of ancient Greece. Unlike Greco-Roman wrestling, which prohibits grabs below the waist, freestyle wrestling allows competitors to take advantage of a greater variety of trips and holds. In a freestyle wrestling match, participants attempt to pin one another by holding the opponent’s shoulders directly to the mat.
Grappling: A popular martial art throughout the world, grappling draws upon many of the same techniques as more conventional forms of wrestling. In particular, grappling places special importance on the use of submission holds, which immobilize opponents and force them to tap out of the fight. Combatants begin matches on their feet, at which point they attempt to take their opponent to the ground in a favorable position. Many martial arts make use of grappling techniques, including Brazilian Jiu-Jitsu, judo, sambo, and various mixed martial arts.
Sambo: From the Russian acronym for “self-defense without weapons,” sambo originated in the Soviet Union during the 1900s. The sport draws upon techniques from various other wrestling styles and martial arts disciplines, including freestyle wrestling, jiu-jitsu, and even folk wrestling.
Beach wrestling: One of the newer wrestling disciplines, beach wrestling made its debut as an official wrestling style in 2004. Like freestyle wrestling, the objective of beach wrestling is either to throw or take an opponent to his or her back.
[Sources: http://en.wikipedia.org/wiki/Wrestling#International_disciplines_.28non-folk_styles.29]